A riemann-hilbert problem in a riemann surface

Acta Mathematica Scientia 2011,31B(6):2233–2246

http://actams.wipm.ac.cn

A RIEMANN-HILBERT PROBLEM IN

A RIEMANN SURFACE∗

Dedicated to Professor Peter D. Lax on the occasion of his 85th birthday

Spyridon Kamvissis

Department of Applied Mathematics, University of Crete, 714 09 Knossos, Greece

E-mail: [email protected]

Abstract One of the inspirations behind Peter Lax’s interest in dispersive integrable

systems, as the small dispersion parameter goes to zero, comes from systems of ODEs

discretizing 1-dimensional compressible gas dynamics [17]. For example, an understanding

of the asymptotic behavior of the Toda lattice in different regimes has been able to shed

light on some of von Neumann’s conjectures concerning the validity of the approximation

of PDEs by dispersive systems of ODEs.

Back in the 1990s several authors have worked on the long time asymptotics of the Toda

lattice [2, 7, 8, 19]. Initially the method used was the method of Lax and Levermore [16],

reducing the asymptotic problem to the solution of a minimization problem with constraints

(an “equilibrium measure” problem). Later, it was found that the asymptotic method of

Deift and Zhou (analysis of the associated Riemann-Hilbert factorization problem in the

complex plane) could apply to previously intractable problems and also produce more

detailed information.

Recently, together with Gerald Teschl, we have revisited the Toda lattice; instead of solu-

tions in a constant or steplike constant background that were considered in the 1990s we

have been able to study solutions in a periodic background.

Two features are worth noting here. First, the associated Riemann-Hilbert factorization

problem naturally lies in a hyperelliptic Riemann surface. We thus generalize the Deift-

Zhou “nonlinear stationary phase method” to surfaces of nonzero genus. Second, we illus-

trate the important fact that very often even when applying the powerful Riemann-Hilbert

method, a Lax-Levermore problem is still underlying and understanding it is crucial in the

analysis and the proofs of the Deift-Zhou method!

Key words Riemann-Hilbert problem; Toda lattice

2000 MR Subject Classification 37K40; 37K45; 35Q15; 37K10

∗Received August 17, 2011. Research supported in part by the ESF program MISGAM and the EU program

ACMAC at the University of Crete.

2234 ACTA MATHEMATICA SCIENTIA Vol.31 Ser.B

1 Introduction. A Discretization of a System from Gas Dynamics

Consider the following system of 1-dimensional compressible gas dynamics.

ut + px = 0,

Vt − ux = 0, (1.1)

p = p(V ); p′(V ) < 0.

Here u, p, V denote velocity, pressure and specific volume (reciprocal of the density) respectively.

These equations express the conservation of momentum, conservation of mass and the equation

of state.

Consider also the following semi-discretization:

d

dtuk +

pk+1/2 − pk−1/2

Δ= 0,

d

dtVk+1/2 − uk+1 − uk

Δ= 0,

(1.2)

where pk±1/2 = p(Vk±1/2).

Let Xk be chosen such thatdXk

dt = uk and at t = 0,

Xk+1 −Xk

Δ= Vk+1/2. (1.3)

Note that ddt

Xk+1−Xk

Δ = ddtVk+1/2. So, (1.3) holds for all time.

We end up with

d2

dt2+p(

Xk+1−Xk

Δ )− p(Xk−Xk−1

Δ )

Δ= 0. (1.4)

This is a discretization of the hyperbolic equation

Xtt + [p(Xx)]x = 0. (1.5)

As is well known hyperbolic equations suffer a shock at a certain finite time. A natural

question (asked by von Neumann in his study of dispersive schemes) is the following: after the

appearance of violent oscillations (known numerically to be of frequency O(1/Δ)), is there a

weak limit of the solution to the dispersive ODE system? If yes, does the weak limit satisfy the

original equations?

It is now known, thanks to the detailed analysis of the 1990s [19], [7], which was based

on [16], that, at least in the case of the Toda lattice p(V ) = e−V , weak limits exist but do not

satisfy the original equation.

In this article, we show how to extend the analysis of the Toda lattice when the background

is periodic.

2 Long Time Asymptotics of the Periodic Toda Lattice under Short-

Range Perturbations and the Riemann-Hilbert Method

Consider the doubly infinite Toda lattice in Flaschka’s variables

b(n, t) = 2(a(n, t)2 − a(n− 1, t)2),

a(n, t) = a(n, t)(b(n+ 1, t)− b(n, t)),(2.1)

No.6 S. Kamvissis: A RIEMANN-HILBERT PROBLEM IN A RIEMANN SURFACE 2235

(n, t) ∈ Z× R, where the dot denotes differentiation with respect to time.

In the case where one has a constant background (same on both left and right infinity) the

long-time asymptotics were first computed by Novokshenov and Habibullin in the 1980s and

later made rigorous by the author in 1993 [8], under the additional assumption that no solitons

are present. (The full case (with solitons) was only recently presented by Kruger and Teschl).

Here we will consider a quasi-periodic algebro-geometric background solution (aq, bq), to

be described in the next section, plus a short-range perturbation (a, b) satisfying∑n∈Z

n6(|a(n, t)− aq(n, t)|+ |b(n, t)− bq(n, t)|) <∞ (2.2)

for t = 0 and hence for all t ∈ R. The perturbed solution can be computed via the inverse

scattering transform. The case where (aq, bq) is constant is classical while the more general case

we want here was solved only recently in [5].

To fix our background solution, consider a hyperelliptic Riemann surface of genus g with

real moduli E0, E1, · · · , E2g+1. Choose a Dirichlet divisor Dμ and introduce

z(n, t) = Ap0(∞+)− αp0

(Dμ)− nA∞−(∞+) + tU0 − Ξp0∈ C

g, (2.3)

where Ap0(αp0

) is Abel’s map (for divisors) and Ξp0, U0 are some constants defined in the

Appendix. Then our background solution is given in terms of Riemann theta functions by

aq(n, t)2 = a2

θ(z(n+ 1, t))θ(z(n− 1, t))

θ(z(n, t))2,

bq(n, t) = b+1

2

d

dtlog

(θ(z(n, t))

θ(z(n− 1, t))

),

(2.4)

where a, b ∈ R are again some constants.

We can of course view this hyperelliptic Riemann surface as formed by cutting and pasting

two copies of the complex plane along bands. Having this picture in mind, we denote the

standard projection to the complex plane by π.

Assume for simplicity that the Jacobi operator

H(t)f(n) = a(n, t)f(n+ 1) + a(n− 1, t)f(n− 1) + b(n, t)f(n), f ∈ �2(Z), (2.5)

corresponding to the perturbed problem (2.1) has no eigenvalues. Then, for long times the

perturbed Toda lattice is asymptotically close to the following limiting lattice defined by

∞∏j=n

(al(j, t)

aq(j, t)

)2=

θ(z(n, t))

θ(z(n− 1, t))

θ(z(n− 1, t) + δ(n, t))

θ(z(n, t) + δ(n, t))

× exp(

1

2πi

∫C(n/t)

log(1− |R|2)ω∞+∞−

), (2.6)

δ�(n, t) =1

2πi

∫C(n/t)

log(1− |R|2)ζ�,

where R is the reflection coefficient defined when considering scattering with respect to the

periodic background (see the second appendix), ζ� is a canonical basis of holomorphic differ-

entials, ω∞+∞− is an Abelian differential of the third kind defined in (A.15), and C(n/t) is a


contour on the Riemann surface. More specific, C(n/t) is obtained by taking the spectrum of

the unperturbed Jacobi operator Hq between −∞ and a special stationary phase point zj(n/t),

for the phase of the underlying Riemann-Hilbert problem defined in the appendix, and lifting it

to the Riemann surface (oriented such that the upper sheet lies to its left). The point zj(n/t)

will move from −∞ to +∞ as n/t varies from −∞ to +∞. From the products above, one easily

recovers al(n, t). More precisely, we have the following.

Theorem 2.1 Let C be any (large) positive number and δ be any (small) positive num-

ber. Consider the region D = {(n, t) : |nt | < C}. Then one has∞∏

j=n

al(j, t)

a(j, t)→ 1 (2.7)

uniformly in D, as t→∞.

Fig. 1 Numerically computed solution of the Toda lattice, with initial condition

a period two solution perturbed at one point in the middle.

Remark 2.2 (i) A naive guess would be that the perturbed periodic lattice approaches

the unperturbed one in the uniform norm. However, this is wrong: In Figure 1 the two observed

lines express the variables a(n, t) of the Toda lattice (see (2.1) below) at a frozen (fairly large)

time t. In areas where the lines seem to be continuous this is due to the fact that we have

plotted a huge number of particles and also due to the 2-periodicity in space. So one can think

of the two lines as the even- and odd-numbered particles of the lattice. We first note the single

soliton which separates two regions of apparent periodicity on the left. Also, after the soliton,

we observe three different areas with apparently periodic solutions of period two. Finally there

are some transitional regions in between which interpolate between the different period two

regions. The theorem above gives a rigorous and complete mathematical explanation of this

picture.

(ii) It is easy to see how the asymptotic formula above describes the picture given by the

numerics. Recall that the spectrum σ(Hq) of Hq consists of g + 1 bands whose band edges

are the branch points of the underlying hyperelliptic Riemann surface. If nt is small enough,

zj(n/t) is to the left of all bands implying that C(n/t) is empty and thus δ�(n, t) = 0; so we

recover the purely periodic lattice. At some value of nt a stationary phase point first appears

in the first band of σ(Hq) and begins to move form the left endpoint of the band towards the

right endpoint of the band. (More precisely we have a pair of stationary phase points zj and

z∗j , one in each sheet of the hyperelliptic curve, with common projection π(zj) on the complex


plane.) So δ�(n, t) is now a non-zero quantity changing with nt and the asymptotic lattice has

a slowly modulated non-zero phase. Also the factor given by the exponential of the integral is

non-trivially changing with nt and contributes to a slowly modulated amplitude. Then, after the

stationary phase point leaves the first band there is a range of nt for which no stationary phase

point appears in the spectrum σ(Hq), hence the phase shift δ�(n, t) and the integral remain

constant, so the asymptotic lattice is periodic (but with a non-zero phase shift). Eventually a

stationary phase point appears in the second band, so a new modulation appears and so on.

Finally, when nt is large enough, so that all bands have been traversed by the stationary phase

point(s), the asymptotic lattice is again periodic. Periodicity properties of theta functions easily

show that phase shift is actually cancelled by the exponential of the integral and we recover

the original periodic lattice with no phase shift at all.

(iii) If eigenvalues are present one can apply appropriate Darboux transformations to add

the effect of such eigenvalues. Alternatively one can modify the Riemann-Hilbert by adding

small circles around the extra poles coming from the eigenvalues and applying some of the

methods in [2]. What we then see asymptotically is travelling solitons in a periodic background.

Note that this will change the asymptotics on one side. See [12] for the exact statement and

proof.

(iv) It is very easy to also show that in any region |nt | > C, one has

∞∏j=n

al(j, t)

a(j, t)→ 1 (2.8)

uniformly in t, as n→∞.

By dividing in (2.6) one recovers the a(n, t). It follows from the theorem above that

|a(n, t)− al(n, t)| → 0 (2.9)

uniformly in D, as t → ∞. In other words, the perturbed Toda lattice is asymptotically close

to the limiting lattice above.

A similar theorem can be proved for the velocities b(n, t).

Theorem 2.3 In the region D = {(n, t) : |nt | < C}, of Theorem 2.1 we also have

∞∑j=n

(bl(j, t)− bq(j, t)

)→ 0 (2.10)

uniformly in D, as t→∞, where bl is given by

∞∑j=n

(bl(j, t) − bq(j, t)

)

=1

2πi

∫C(n/t)

log(1− |R|2)Ω0 + 1

2

d

dslog

(θ(z(n, s) + δ(n, t))

θ(z(n, s))

) ∣∣∣∣s=t

(2.11)

and Ω0 is an Abelian differential of the second kind defined in (A.16).

The next question we address here concerns the higher order asymptotics. Namely, what

is the rate at which the perturbed lattice approaches the limiting lattice? Even more, what is

the exact asymptotic formula? The answer is given by


Theorem 2.4 Let Dj be the sector Dj = {(n, t), : zj(n/t) ∈ [E2j+ε, E2j+1−ε] for someε > 0. Then one has

∞∏j=n

(a(j, t)

al(j, t)

)2= 1 +

√i

φ′′(zj(n/t))t2Re

(β(n, t)iΛ0(n, t)

)+O(t−α) (2.12)

and

∞∑j=n+1

(b(j, t)− bl(j, t)

)=

√i

φ′′(zj(n/t))t2Re

(β(n, t)iΛ1(n, t)

)+O(t−α) (2.13)

for any α < 1 uniformly in Dj , as t→∞. Here

φ′′(zj)/i =

g∏k=0,k �=j

(zj − zk)

iR1/22g+2(zj)

> 0, (2.14)

(where φ(p, n/t) is the phase function defined in (B.17) and R1/22g+2(z) the square root of the

underlying Riemann surface),

Λ0(n, t) = ω∞−∞+(zj) +

∑k,�

ck�(ν(n, t))

∫ ∞−

∞+

ων�(n,t),0ζk(zj),

Λ1(n, t) = ω∞−,0(zj)−∑k,�

ck�(ν(n, t))ων�(n,t),0(∞+)ζk(zj), (2.15)

with ck�(ν(n, t)) some constants defined by

(c�k(ν))1≤�,k≤g =

⎛⎝ g∑

j=1

ck(j)μj−1

� dπ

R1/22g+2(μ�)

⎞⎠−1

1≤�,k≤g

(2.16)

where ck(j) are defined in (A.6), ωq,0 an Abelian differential of the second kind with a second

order pole at q,

β =√νei(π/4−arg(R(zj)))+arg(Γ(iν))−2να(zj))

(φ′′(zj)

i

)iνe−tφ(zj)t−iν

×θ(z(zj , n, t) + δ(n, t))

θ(z(zj, 0, 0))

θ(z(z∗j , 0, 0))

θ(z(z∗j , n, t) + δ(n, t))

× exp(

1

2πi

∫C(n/t)

log

(1− |R|2

1− |R(zj)|2)ωp p∗

), (2.17)

where Γ(z) is the gamma function,

ν = − 1

2πlog(1− |R(zj)|2) > 0 (2.18)

and α(zj) is a real constant defined by

1

2

∫C(n/t)

ωp p∗ = ± log(z − zj)± α(zj) +O(z − zj), p = (z,±), (2.19)


where ωp q is the meromorphic differential of the third kind with poles at p, q with residues

1,−1 respectively.The last theorem and its proof have the following interpretation: even when a Riemann-

Hilbert problem needs to be considered on an algebraic variety, a localized parametrix Riemann-

Hilbert problem need only be solved in the complex plane and the local solution can then be

glued to the global Riemann-Hilbert solution on the variety.

Similarly one can study the local higher asymptotics in the small (decaying actually) “res-

onance” regions that we excluded in the last theorem above. There is a Painleve region where

the higher order asymptotics are given in terms of a solution of Painleve II and a “collision-

less shock” region where the higher order asymptotics are given in terms of the elliptic cosine

function cn.

The proof of all three theorems is given in [15]. It is a stationary phase type argument.

One reduces the given Riemann-Hilbert problem to a localized parametrix Riemann-Hilbert

problem. This is done via the solution of a scalar global Riemann-Hilbert problem which is

solved explicitly with the help of the Riemann-Roch theorem. The reduction to a localized

parametrix Riemann-Hilbert problem is done with the help of a theorem reducing general

Riemann-Hilbert problems to singular integral equations. (A generalized Cauchy transform is

defined appropriately for each Riemann surface.) The localized parametrix Riemann-Hilbert

problem is solved explicitly in terms of parabolic cylinder functions. The argument follows [3]

up to a point but also extends the theory of Riemann-Hilbert problems for Riemann surfaces.

The right (well-posed) Riemann-Hilbert factorization problems are no more holomorphic but

instead have a number of poles equal to the genus of the surface.

3 The Generalized Toda Shock Problem and the Return of Lax-

Levermore

Suppose now that we have different backgrounds at the infinite ends of the lattice. This

could mean different genus or even same genus but different isospectral class. What happens

then? This is the subject of an ongoing investigation wih Gerald Teschl and our theorem will

not be stated in full detail here. We will only note a few things.

First, as in the previous case, the n, t-plane is also divided in several regions. There are

“modulated” regions and “periodic” regions (which actually become “soliton” regions when

eigenvalues are allowed).

The main difference is that not all regions are associated to Riemann surfaces of the same

genus. They couldn’t, since by assumption we have data of different genus at the infinite ends

of the lattice. So, as n/t ranges from −∞ to +∞ the moduli of the underlying Riemann surface

change, but also there are singular pinching points where there is a change in genus.

On the level of the Riemann-Hilbert analysis, the underlying Riemann surface can be

chosen to have genus compatible with either (left or right) asymptotics of the initial data. But

now there is an extra issue: instead of the constant band/gap structure of the Riemann surface,

there is also an additional n/t dependent band/gap structure coming from a Lax-Levermore

type variational problem. The support of the minimizer is a finite union of bands!

As already noted in [4] and numerous publications in the application of the Riemann-


Hilbert theory to orthogonal polynomials and random matrices (see [1] for a clear detaied

exposition), when one needs to use the full power of the Deift-Zhou method an underlying

Lax-Levermore problem has to be understood. In fact, a minimizer (or in some cases a maxi-

minimizer [10, 11, 13]) has to be constructed!1 In our case, the Lax-Levermore minimizer lives

on a curve in a hyperelliptic surface. Conjugation by the log-transform of the minimizer enables

us to transform our given Riemann-Hilbert problem to one that is explicitly solvable.

Acknowledgments I gratefully acknowledge the support of the European Science Foun-

dation (MISGAM program) and the support of the European Commission through the ACMAC

program of the Department of Applied Mathematics in Crete.

References

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1While the original Lax-Levermore problem was defined on the real line, the maxi-min extension is in general

defined in the complex plane or even in an infinite-sheeted Riemann surface.


Appendix A Algebro-geometric Quasi-periodic Finite-gap Solutions

We present some facts on our background solution (aq, bq) which we want to choose from

the class of algebro-geometric quasi-periodic finite-gap solutions, that is the class of stationary

solutions of the Toda hierarchy. In particular, this class contains all periodic solutions. We will

use the same notation as in [18], where we also refer to for proofs.

To set the stage let M be the Riemann surface associated with the following function

R1/22g+2(z), R2g+2(z) =

2g+1∏j=0

(z − Ej), E0 < E1 < · · · < E2g+1, (A.1)

g ∈ N. M is a compact, hyperelliptic Riemann surface of genus g. We will choose R1/22g+2(z) as

the fixed branch

R1/22g+2(z) = −

2g+1∏j=0

√z − Ej , (A.2)

where√. is the standard root with branch cut along (−∞, 0).

A point on M is denoted by p = (z,±R1/22g+2(z)) = (z,±), z ∈ C, or p = (∞,±) = ∞±,

and the projection onto C ∪ {∞} by π(p) = z. The points {(Ej , 0), 0 ≤ j ≤ 2g + 1} ⊆ M are

called branch points and the sets

Π± = {(z,±R1/22g+2(z)) | z ∈ C \g⋃

j=0

[E2j , E2j+1]} ⊂ M (A.3)

are called upper, lower sheet, respectively.

Let {aj, bj}gj=1 be loops on the surface M representing the canonical generators of the

fundamental group π1(M). We require aj to surround the points E2j−1, E2j (thereby changing

sheets twice) and bj to surround E0, E2j−1 counterclockwise on the upper sheet, with pairwise

intersection indices given by

ai ◦ aj = bi ◦ bj = 0, ai ◦ bj = δi,j , 1 ≤ i, j ≤ g. (A.4)

The corresponding canonical basis {ζj}gj=1 for the space of holomorphic differentials can be

constructed by

ζ =

g∑j=1

c(j)πj−1dπ

R1/22g+2

, (A.5)

where the constants c(.) are given by

cj(k) = C−1jk , Cjk =

∫ak

πj−1dπ

R1/22g+2

= 2

∫ E2k

E2k−1

zj−1dz

R1/22g+2(z)

∈ R. (A.6)

The differentials fulfill∫aj

ζk = δj,k,

∫bj

ζk = τj,k, τj,k = τk,j , 1 ≤ j, k ≤ g. (A.7)

Now pick g numbers (the Dirichlet eigenvalues)

(μj)gj=1 = (μj , σj)

gj=1 (A.8)


whose projections lie in the spectral gaps, that is, μj ∈ [E2j−1, E2j ]. Associated with these

numbers is the divisor Dμ which is one at the points μj and zero else. Using this divisor we

introduce

z(p, n, t) = Ap0(p)− αp0

(Dμ)− nA∞−(∞+) + tU0 − Ξp0∈ Cg,

z(n, t) = z(∞+, n, t), (A.9)

where Ξp0is the vector of Riemann constants

Ξp0,j =

j +g∑

k=1

τj,k

2, p0 = (E0, 0), (A.10)

U0 are the b-periods of the Abelian differential Ω0 defined below, and Ap0(αp0

) is Abel’s

map (for divisors). The hat indicates that we regard it as a (single-valued) map from M (the

fundamental polygon associated with M by cutting along the a and b cycles) to Cg. We recall

that the function θ(z(p, n, t)) has precisely g zeros μj(n, t) (with μj(0, 0) = μj), where θ(z) is

the Riemann theta function of M.

Then our background solution is given by

aq(n, t)2 = a2

θ(z(n+ 1, t))θ(z(n− 1, t))

θ(z(n, t))2,

bq(n, t) = b+1

2

d

dtlog( θ(z(n, t))

θ(z(n− 1, t))

). (A.11)

The constants a, b depend only on the Riemann surface (see [18, Section 9.2]).

Introduce the time dependent Baker-Akhiezer function

ψq(p, n, t) = C(n, 0, t)θ(z(p, n, t))

θ(z(p, 0, 0))exp

(n

∫ p

E0

ω∞+∞− + t

∫ p

E0

Ω0

), (A.12)

where C(n, 0, t) is real-valued,

C(n, 0, t)2 =θ(z(0, 0))θ(z(−1, 0))θ(z(n, t))θ(z(n− 1, t))

, (A.13)

and the sign has to be chosen in accordance with aq(n, t). Here

θ(z) =∑

m∈Zg

exp 2πi

(〈m, z〉+ 〈m, τ m〉

2

), z ∈ C

g, (A.14)

is the Riemann theta function associated with M,

ω∞+∞− =

g∏j=1

(π − λj)

R1/22g+2

dπ (A.15)

is the Abelian differential of the third kind with poles at ∞+ and ∞− and

Ω0 =

g∏j=0

(π − λj)

R1/22g+2

dπ,

g∑j=0

λj =1

2

2g+1∑j=0

Ej , (A.16)


is the Abelian differential of the second kind with second order poles at∞+ respectively∞− (see

[18, Sects. 13.1, 13.2]). All Abelian differentials are normalized to have vanishing aj periods.

The Baker-Akhiezer function is a meromorphic function on M \ {∞±} with an essential

singularity at ∞±. The two branches are denoted by

ψq,±(z, n, t) = ψq(p, n, t), p = (z,±) (A.17)

and it satisfies

Hq(t)ψq(p, n, t) = π(p)ψq(p, n, t),

d

dtψq(p, n, t) = Pq,2(t)ψq(p, n, t), (A.18)

where

Hq(t)f(n) = aq(n, t)f(n+ 1) + aq(n− 1, t)f(n− 1) + bq(n, t)f(n), (A.19)

Pq,2(t)f(n) = aq(n, t)f(n+ 1)− aq(n− 1, t)f(n− 1) (A.20)

are the operators from the Lax pair for the Toda lattice.

It is well known that the spectrum of Hq(t) is time independent and consists of g+1 bands

σ(Hq) =

g⋃j=0

[E2j , E2j+1]. (A.21)

For further information and proofs we refer to [18, Chap. 9 and Sect. 13.2].

Appendix B The Inverse Scattering Transform and the Riemann-

Hilbert Problem

In this section our notation and results are taken from [5] and [6]. Let ψq,±(z, n, t) be the

branches of the Baker-Akhiezer function defined in the previous section. Let ψ±(z, n, t) be the

Jost functions for the perturbed problem

a(n, t)ψ±(z, n+ 1, t) + a(n− 1, t)ψ±(z, n− 1, t) + b(n, t)ψ±(z, n, t) = zψ±(z, n, t) (B.1)

defined by the asymptotic normalization

limn→±∞

w(z)∓n(ψ±(z, n, t)− ψq,±(z, n, t)) = 0, (B.2)

where w(z) is the quasimomentum map

w(z) = exp(

∫ p

E0

ω∞+∞−), p = (z,+). (B.3)

The asymptotics of the two projections of the Jost function are

ψ±(z, n, t) = ψq,±(z, 0, t)

z∓n( n−1∏

j=0

aq(j, t))±1

A±(n, t)

×(1 +

(B±(n, t)±

n∑j=1

bq(j − 0

1, t))1z+O(

1

z2)), (B.4)


as z →∞, where

A+(n, t) =

∞∏j=n

a(j, t)

aq(j, t), B+(n, t) =

∞∑j=n+1

(bq(j, t) − b(j, t)),

A−(n, t) =

n−1∏j=−∞

a(j, t)

aq(j, t), B−(n, t) =

n−1∑j=−∞

(bq(j, t) − b(j, t)).(B.5)

One has the scattering relations

T (z)ψ∓(z, n, t) = ψ±(z, n, t) +R±(z)ψ±(z, n, t), z ∈ σ(Hq), (B.6)

where T (z), R±(z) are the transmission respectively reflection coefficients. Here ψ±(z, n, t) is

defined such that ψ±(z, n, t) = limε↓0

ψ±(z + iε, n, t), z ∈ σ(Hq). If we take the limit from the

other side we have ψ±(z, n, t) = limε↓0

ψ±(z − iε, n, t).

The transmission T (z) and reflection R±(z) coefficients satisfy

T (z)R+(z) + T (z)R−(z) = 0, |T (z)|2 + |R±(z)|2 = 1. (B.7)

In particular one reflection coefficient, say R(z) = R+(z), suffices.

We will define a Riemann-Hilbert problem on the Riemann surface M as follows:

m(p, n, t) =

⎧⎨⎩T (z)ψ−(z, n, t) ψ+(z, n, t), p = (z,+),

ψ+(z, n, t) T (z)ψ−(z, n, t), p = (z,−).(B.8)

Note that m(p, n, t) inherits the poles at μj(0, 0) and the essential singularity at ∞± from the

Baker–Akhiezer function.

We are interested in the jump condition of m(p, n, t) on Σ, the boundary of Π± (oriented

counterclockwise when viewed from top sheet Π+). It consists of two copies Σ± of σ(Hq)

which correspond to non-tangential limits from p = (z,+) with ±Im(z) > 0, respectively to

non-tangential limits from p = (z,−) with ∓Im(z) > 0.

To formulate our jump condition we use the following convention: When representing

functions on Σ, the lower subscript denotes the non-tangential limit from Π+ or Π−, respectively,

m±(p0) = limΠ±�p→p0

m(p), p0 ∈ Σ. (B.9)

Using the notation above implicitly assumes that these limits exist in the sense that m(p)

extends to a continuous function on the boundary away from the band edges.

Moreover, we will also use symmetries with respect to the sheet exchange map

p∗ =

⎧⎨⎩ (z,∓) for p = (z,±),∞∓ for p =∞±,

(B.10)

and complex conjugation

p =

⎧⎪⎪⎨⎪⎪⎩(z,±) for p = (z,±) �∈ Σ,(z,∓) for p = (z,±) ∈ Σ,∞± for p =∞±.

(B.11)


In particular, we have p = p∗ for p ∈ Σ.Note that we have m±(p) = m∓(p

∗) for m(p) = m(p∗) (since ∗ reverses the orientation ofΣ) and m±(p) = m±(p∗) for m(p) = m(p).

With this notation, using (B.6) and (B.7), we obtain

m+(p, n, t) = m−(p, n, t)

⎛⎝ |T (p)|2 −R(p)

R(p) 1

⎞⎠ , (B.12)

where we have extended our definition of T to Σ such that it is equal to T (z) on Σ+ and equal

to T (z) on Σ−. Similarly for R(z). In particular, the condition on Σ+ is just the complex

conjugate of the one on Σ− since we have R(p∗) = R(p) and m±(p∗, n, t) = m±(p, n, t) for

p ∈ Σ.To remove the essential singularity at ∞± and to get a meromorphic Riemann-Hilbert

problem we set

m2(p, n, t) = m(p, n, t)

⎛⎝ψq(p

∗, n, t)−1 0

0 ψq(p, n, t)−1

⎞⎠ . (B.13)

Its divisor satisfies

(m21) ≥ −Dμ(n,t)∗ , (m2

2) ≥ −Dμ(n,t), (B.14)

and the jump conditions become

m2+(p, n, t) = m2

−(p, n, t)J2(p, n, t),

J2(p, n, t) =

⎛⎝ 1− |R(p)|2 −R(p)Θ(p, n, t)e−tφ(p)

R(p)Θ(p, n, t)etφ(p) 1

⎞⎠ , (B.15)

where

Θ(p, n, t) =θ(z(p, n, t))

θ(z(p, 0, 0))

θ(z(p∗, 0, 0))

θ(z(p∗, n, t))(B.16)

and

φ(p,n

t) = 2

∫ p

E0

Ω0 + 2n

t

∫ p

E0

ω∞+∞− ∈ iR (B.17)

for p ∈ Σ. Noteψq(p, n, t)

ψq(p∗, n, t)= Θ(p, n, t)etφ(p).

Observe that

m2(p) = m2(p)

and

m2(p∗) = m2(p)

(0 1

1 0

),

which follow directly from the definition (B.13). They are related to the symmetries

J2(p) = J2(p) and J2(p) =

(0 1

1 0

)J2(p∗)−1

(0 1

1 0

).


Now we come to the normalization condition at ∞+. To this end note

m(p, n, t) =

(A+(n, t)(1 −B+(n− 1, t)

1

z)

1

A+(n, t)(1 +B+(n, t)

1

z)

)+O(

1

z2), (B.18)

for p = (z,+) → ∞+, with A±(n, t) and B±(n, t) are defined in (B.5). The formula near ∞−

follows by flipping the columns. Here we have used

T (z) = A−(n, t)A+(n, t)

(1− B+(n, t) + bq(n, t)− b(n, t) +B−(n, t)

z+O(

1

z2)

). (B.19)

Using the properties of ψ(p, n, t) and ψq(p, n, t) one checks that its divisor satisfies

(m1) ≥ −Dμ(n,t)∗ , (m2) ≥ −Dμ(n,t). (B.20)

Next we show how to normalize the problem at infinity. The use of the above symmetries is

necessary and it makes essential use of the second sheet of the Riemann surface.

Theorem B.1 The function

m3(p) =1

A+(n, t)m2(p, n, t) (B.21)

with m2(p, n, t) defined in (B.13) is meromorphic away from Σ and satisfies:

m3+(p) = m3

−(p)J3(p), p ∈ Σ,

(m31) ≥ −Dμ(n,t)∗ , (m3

2) ≥ −Dμ(n,t), (B.22)

m3(p∗) = m3(p)

(0 1

1 0

),

m3(∞+) =(1 ∗), (B.23)

where the jump is given by

J3(p, n, t) =

⎛⎝ 1− |R(p)|2 −R(p)Θ(p, n, t)e−tφ(p)

R(p)Θ(p, n, t)etφ(p) 1

⎞⎠ . (B.24)

Setting R(z) ≡ 0 we clearly recover the purely periodic solution, as we should. Moreover,

note

m3(p) =

(1

A+(n, t)21

)+

(B+(n, t)

A+(n, t)2−B+(n− 1, t)

)1

z+O(

1

z2). (B.25)

for p = (z,−) near ∞−.

Existence of a solution of the normalized Riemann-Hilbert problem follows by construction;

for uniqueness see [15].

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A riemann-hilbert problem in a riemann surface